Asymptotic behavior of flat surfaces in hyperbolic 3-space
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- KOKUBU Masatoshi
- Department of Mathematics, School of Engineering, Tokyo Denki University
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- ROSSMAN Wayne
- Department of Mathematics, Faculty of Science, Kobe University
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- UMEHARA Masaaki
- Department of Mathematics, Graduate School of Science, Osaka University
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- YAMADA Kotaro
- Faculty of Mathematics, Kyushu University
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説明
In this paper, we investigate the asymptotic behavior of regular ends of flat surfaces in the hyperbolic 3-space H3. Gálvez, Martínez and Milán showed that when the singular set does not accumulate at an end, the end is asymptotic to a rotationally symmetric flat surface. As a refinement of their result, we show that the asymptotic order (called pitch p) of the end determines the limiting shape, even when the singular set does accumulate at the end. If the singular set is bounded away from the end, we have −1<p≤0. If the singular set accumulates at the end, the pitch p is a positive rational number not equal to 1. Choosing appropriate positive integers n and m so that p=n⁄m, suitable slices of the end by horospheres are asymptotic to d-coverings (d-times wrapped coverings) of epicycloids or d-coverings of hypocycloids with 2n0 cusps and whose normal directions have winding number m0, where n=n0d, m=m0d (n0, m0 are integers or half-integers) and d is the greatest common divisor of m−n and m+n. Furthermore, it is known that the caustics of flat surfaces are also flat. So, as an application, we give a useful explicit formula for the pitch of ends of caustics of complete flat fronts.
収録刊行物
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- Journal of the Mathematical Society of Japan
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Journal of the Mathematical Society of Japan 61 (3), 799-852, 2009
一般社団法人 日本数学会
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詳細情報 詳細情報について
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- CRID
- 1390001205116607360
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- NII論文ID
- 10026998469
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- NII書誌ID
- AA0070177X
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- ISSN
- 18811167
- 18812333
- 00255645
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- NDL書誌ID
- 10286803
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- 本文言語コード
- en
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- 資料種別
- journal article
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